Sophisticated data collection facilities often produce data which are a set of functions, represented in the form of curves, images or shapes. The development of functional data analysis in theory and methodology has provided us important analytical tools to address challenging problems encountered in many important fields. In the proposed research, the investigator continues to build and enrich the theory and methodology of functional data. This proposal targets the development of powerful statistical tools for analyzing functional data in time series and survey sampling frameworks. Four related research topics are proposed for investigation. For each project, statistical properties of the estimators, statistical inferences governed by the underlying models, and theoretical properties of the inferences will be studied. The proposed methods can be used to estimate global quantities for dependent functional data, quantify and visualize the variability of the estimators, and make global inferences on the shape of the population quantities.
With "big data" of complex (such as longitudinal, functional, heterogeneous, or correlated) features becoming increasingly available for public use in many research areas, this proposal is one vehicle to address the challenges of analyzing such types of data. The success of the proposed projects provides effective and practical tools for dealing with large and complex structural data over time and space, representing advances in the theory and methodology of statistical analysis. The benefits to society at large of the proposed research include new methodology and inference tools for big data with complex features. These topics are of interest to statisticians, survey researchers, and indeed, more broadly for researchers in climatology, health, economics, engineering, environmental studies, meteorology, behavioral and social sciences.
